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Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces

Azagra Rueda, Daniel and Fabián, M. and Jiménez Sevilla, María del Mar (2005) Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces. Canadian Mathematical Bulletin, 48 (4). pp. 481-499. ISSN 0008-4395

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Abstract

We establish sufficient conditions on the shape of a set A included in
the space Ln s (X; Y ) of the n-linear symmetric mappings between Banach spaces
X and Y , to ensure the existence of a Cn-smooth mapping f : X ¡! Y , with bounded support, and such that f(n)(X) = A, provided that X admits a Cn- smooth bump with bounded n-th derivative and densX = densLn(X; Y ). For instance, when X is infinite-dimensional, every bounded connected and open set U containing the origin is the range of the n-th derivative of such a mapping.
The same holds true for the closure of U, provided that every point in the boundary of U is the end point of a path within U. In the finite-dimensional case, more restrictive conditions are required. We also study the Fr´echet smooth case for mappings from Rn to a separable infinite-dimensional Banach space and the Gˆateaux smooth case for mappings defined on a separable infinite-dimensional
Banach space and with values in a separable Banach space.


Item Type:Article
Uncontrolled Keywords:Starlike Bodies; Range; Theorem; Bump
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:12927
Deposited On:11 Jul 2011 08:06
Last Modified:06 Feb 2014 09:36

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